A negative scalar projection means that a vector’s component along a chosen axis points opposite to the axis’s positive direction. To calculate the signed component of a along b, take a · b and divide by the magnitude of b, provided b is non-zero.
The 2027 A-Level H2 Mathematics 9758 syllabus covers the geometric meaning of the scalar product with a unit vector. This guide distinguishes the signed component, the projection vector and the length of that vector.
- Normalise the direction vector before taking the scalar product.
- Keep the sign when the question asks for a scalar component.
- Multiply the signed component by the unit direction for the projection vector.
- Take the magnitude if the question asks for length.
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Set the positive direction first
A unit vector specifies a direction with magnitude 1. If b is a non-zero direction vector, u = b/|b| is the unit vector pointing the same way.
The scalar component along u is a · u. It equals |a| cos θ when a is non-zero and θ is the angle between a and u. An obtuse angle gives a negative component.
The vector projection is (a · u)u. It lies along the axis and points the way the sign indicates.
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Worked example | Project (2, −6, 1) onto the direction (1, 1, 0)
The direction vector has magnitude √2, so its unit vector is (1, 1, 0)/√2. The scalar product of a with b is 2 − 6 = −4.
| Asked for | Result |
|---|---|
| Signed component along b | −4/√2 = −2√2 |
| Projection vector | (−2, −2, 0) |
| Length of projection | 2√2 |
The negative component means the projection vector points opposite to (1, 1, 0). The vector left after removing the projection is (4, −4, 1). Its dot product with b is 4 − 4 = 0, so it is perpendicular to the axis.
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Practise by reversing the direction
Replace b with (−1, −1, 0). The scalar product with a becomes 4 and the magnitude of the direction stays √2. The signed component is now +2√2.
The projection vector is still (−2, −2, 0). The positive direction reversed and the signed component reversed with it, so their product is unchanged.
If the question asks for the distance from a point with position vector a to the line through the origin in the direction b, use the perpendicular part. Here its magnitude is √33.
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Frequently asked questions
What if b is already a unit vector?
The scalar component is just a · b, since |b| = 1.
Can I project onto the zero vector?
The zero vector specifies no direction and cannot be normalised.
What if the scalar product is zero?
The projected component is zero. If both original vectors are non-zero, they are perpendicular.
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